Question

Equation of two diameters of a circle are 2 x − 3 y = 5 and 3 x − 4 y = 7. The line joining the points ( − 22 7 , − 4 ) and ( − 1 7 , 3 ) intersects the circle at only one point P ( α , β ). Then 17 β − α is equal to

• A. 2

Answer 1

Answer: (A)

2

Answer 2

Solution:

1. Find the center:

Since the diameters are given by

$$2x-3y=5 \quad \text{and} \quad 3x-4y=7,$$

their intersection is the center. Solve:

$$x=\frac{5+3y}{2}.$$

Substitute in $3x-4y=7$:

$$3\left(\frac{5+3y}{2}\right)-4y=7 \implies \frac{15+9y}{2}-4y=7.$$

Multiply by 2:

$$15+9y-8y=14 \implies y=-1.$$

Then,

$$x=\frac{5+3(-1)}{2}=\frac{2}{2}=1.$$

Center $C=(1,-1)$.

2. Determine the tangent line:

The line through the points

$$\left(-\frac{22}{7},-4\right) \quad \text{and} \quad \left(-\frac{1}{7},3\right)$$

has slope:

$$m=\frac{3-(-4)}{(-1/7)-(-22/7)}=\frac{7}{21/7}=\frac{7}{3}.$$

Thus its equation (using point-slope form) is:

$$y+4=\frac{7}{3}\Bigl(x+\frac{22}{7}\Bigr) \quad\Longrightarrow\quad y=\frac{7}{3}x+\frac{10}{3}.$$

Since the line meets the circle only at one point $P$, it is tangent.

3. Find the point of tangency $P(\alpha,\beta)$:

For a tangent, the radius $CP$ is perpendicular to the tangent.

• The slope of the tangent is $\frac{7}{3}$, so the slope of $CP$ is $-\frac{3}{7}$.
• The equation of $CP$ is: $$y+1=-\frac{3}{7}(x-1) \quad\Longrightarrow\quad y=-\frac{3}{7}x-\frac{4}{7}.$$

Since $P$ lies on both lines, equate:

$$\frac{7}{3}x+\frac{10}{3}=-\frac{3}{7}x-\frac{4}{7}.$$

Multiply by 21 to clear denominators:

$$49x+70=-9x-12.$$

Solving,

$$58x=-82 \quad\Longrightarrow\quad x=-\frac{41}{29}.$$

Substitute $x$ in the equation of $CP$:

$$y=-\frac{3}{7}\Bigl(-\frac{41}{29}\Bigr)-\frac{4}{7}=\frac{123}{203}-\frac{116}{203}=\frac{7}{203}.$$

Thus, $P\left(-\frac{41}{29},\,\frac{7}{203}\right)$.

4. Compute $17\beta -\alpha$: $$17\beta-\alpha=17\Bigl(\frac{7}{203}\Bigr)-\left(-\frac{41}{29}\right)=\frac{119}{203}+\frac{41}{29}.$$

Since $203=7\cdot29$, write

$$\frac{41}{29}=\frac{287}{203}.$$

Therefore,

$$17\beta-\alpha=\frac{119+287}{203}=\frac{406}{203}=2.$$

Explanation (minimal):

• Found center as intersection of diameters: $C=(1,-1)$.
• Tangent line through the given points: $y=\frac{7}{3}x+\frac{10}{3}$.
• The radius through point of tangency $P$ is perpendicular to the tangent, so its equation is: $y=-\frac{3}{7}x-\frac{4}{7}$.
• Solve the two equations to get $P\left(-\frac{41}{29},\,\frac{7}{203}\right)$.
• Calculate $17\beta-\alpha=2$.