Condition for a line to touch a conic; point of tangency — practice questions

Question 1 · difficulty L2 · understanding

If the length of the latus rectum of a parabola, whose focus is $$(a, a)$$ and the tangent at its vertex is $$x+y=a$$, is 16, then $$|a|$$ is equal to :

• A. $$2 \sqrt{2}$$
• B. $$2 \sqrt{3}$$
• C. $$4 \sqrt{2}$$
• D. 4

Question 2 · difficulty L2 · understanding

The ordinates of the points P and $$\mathrm{Q}$$ on the parabola with focus $$(3,0)$$ and directrix $$x=-3$$ are in the ratio $$3: 1$$. If $$\mathrm{R}(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$\mathrm{P}$$ and $$\mathrm{Q}$$, then $$\frac{\beta^{2}}{\alpha}$$ is equal to _______________.

Question 3 · difficulty L2 · understanding

The line $$12x\cos \theta + 5y\sin \theta = 60$$ is tangent to which of the following curves?

• A. x 2 + y 2 = 169
• B. 144x 2 + 25y 2 = 3600
• C. 25x 2 + 12y 2 = 3600
• D. x 2 + y 2 = 60

Question 4 · difficulty L2 · understanding

Let $P$ be the point on the parabola $y = x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2 + y^2 = 2$ such that the slope of the tangent to the circle at the point $Q$ is $-1$. Let $R$ be the point in the first quadrant lying on the ellipse $x^2 + 4y^2 = 8$ such that the slope of the tangent to the ellipse at the point $R$ is $-\frac{1}{2}$. Then the radius of the circle passing through the points $P, Q$ and $R$ is

• A. $\sqrt{10}$
• B. $\sqrt{5}$
• C. $\sqrt{\dfrac{5}{2}}$
• D. $2\sqrt{5}$

Question 5 · difficulty L3 · understanding

Consider the circle C : $x^2+y^2-6 x-8 y-11=0$. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $x^2+y^2-\alpha x-\beta y-\gamma=0$, then $\alpha+\beta+2 \gamma$ is equal to $\_\_\_\_$ .

Question 6 · difficulty L3 · understanding

Consider a circle $$(x-\alpha)^2+(y-\beta)^2=50$$, where $$\alpha, \beta>0$$. If the circle touches the line $$y+x=0$$ at the point $$P$$, whose distance from the origin is $$4 \sqrt{2}$$, then $$(\alpha+\beta)^2$$ is equal to __________.

Question 7 · difficulty L3 · understanding

A circle with centre (2, 3) and radius 4 intersects the line $$x+y=3$$ at the points P and Q. If the tangents at P and Q intersect at the point $$S(\alpha,\beta)$$, then $$4\alpha-7\beta$$ is equal to ___________.

Question 8 · difficulty L3 · understanding

Points P($$-$$3, 2), Q(9, 10) and R($$\alpha,4$$) lie on a circle C and PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line $$2x-ky=1$$, then k is equal to ____________.

Question 9 · difficulty L3 · understanding

Let $$A B$$ be a chord of length 12 of the circle $$(x-2)^{2}+(y+1)^{2}=\frac{169}{4}$$. If tangents drawn to the circle at points $$A$$ and $$B$$ intersect at the point $$P$$, then five times the distance of point $$P$$ from chord $$A B$$ is equal to __________.

Question 10 · difficulty L3 · understanding

Let the lines $$y + 2x = \sqrt {11} + 7\sqrt 7$$ and $$2y + x = 2\sqrt {11} + 6\sqrt 7$$ be normal to a circle $$C:{(x - h)^2} + {(y - k)^2} = {r^2}$$. If the line $$\sqrt {11} y - 3x = {{5\sqrt {77} } \over 3} + 11$$ is tangent to the circle C, then the value of $${(5h - 8k)^2} + 5{r^2}$$ is equal to __________.