Tangents, normals and loci Quiz

1. Find the tangent to $\frac{x^2}{9}+\frac{y^2}{4}=1$ at $(0,2)$.

• A. $x=0$
• B. $y=2$
• C. $y=-2$
• D. $2x+3y=6$

2. For which values of $c$ is $y=x+c$ tangent to $\frac{x^2}{4}+y^2=1$?

• A. $\pm\sqrt3$
• B. $\pm\sqrt5$
• C. $\pm3$
• D. $\pm1$

3. The line $y=x+c$ and the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$: for which $c$ is the line a tangent?

• A. $\pm\sqrt{13}$
• B. $\pm\sqrt{5}$
• C. No real value of $c$
• D. $\pm1$

4. The tangent to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at $(x_1,y_1)$ has gradient $\frac{b^2x_1}{a^2y_1}$. What is the gradient of the normal?

• A. $\frac{b^2x_1}{a^2y_1}$
• B. $\frac{a^2y_1}{b^2x_1}$
• C. $-\frac{b^2x_1}{a^2y_1}$
• D. $-\frac{a^2y_1}{b^2x_1}$

5. $P$ is twice as far from $O(0,0)$ as it is from $A(3,0)$. What is the locus of $P$?

• A. A circle, centre $(4,0)$, radius $2$
• B. A circle, centre $(3,0)$, radius $2$
• C. The line $x=2$
• D. An ellipse